PARADISE LOST?
stated definition of a set (or class) will show that the 4we" men
would not consider the postulate self-evident if the set JJ con-
sisted, say, of an infinity of non-overlapping line segments. Yet
the postulate seems reasonable enough. Attempts to prove it
have failed. It is of considerable importance in all questions
relating to continuity.
A word as to how this postulate came to be introduced into
mathematics will suggest another of the unsolved problems of
Cantor's theory. A set of distinct, countable things, like all the
bricks in a certain wall, can easily be ordered-, we need only
count them off 1, 2, 3, ... in any of dozens of different ways
that will suggest themselves. But how would we go about
ordering all the points on a straight line? They cannot be
counted off 1, 29 3, ... The task appears hopeless when we
consider that between any two points of the line 'we can find',
or 'there exists' another point of the line. If every time we
counted two adjacent bricks another sprang into "being between
them in the wall OUT counting would become slightly confused.
Nevertheless the points on a straight line do appear to have
some sort of order; we can say whether one point is to the right
or the left of another, and so on. Attempts to order the points
of a line have not succeeded. Zennelo proposed his postulate as
a means for making the attempt easier, but it itself is not
universally accepted as a reasonable assumption or as one
which it is safe to use.
Cantor's theory contains a great deal more about the actual
infinite and the 'arithmetic* of transfinite (infinite) numbers
than what has been indicated here. But as the theory is still in
the controversial stage, we may leave it with the statement of
a last riddle. Does there 'exist', or can we 'construct', an infinite
set which is not similar (technical sense of one-to-one matching)
either to the set of all the positive rational integers or to the set
of all points of a line? The answer is unknown.
Cantor died in a mental hospital hi Halle on 6 January 191S
at the age of seventy-three. Honours and recognition were his
at the last, and even the old bitterness against Kronecker was
forgotten. It was no doubt a satisfaction to Cantor to recall that
he and Kronecker had become at least superficially reconciled
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